#94535

order	512
nilpotency class	6
isoclinism	isoclinic class of #94532
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7344
commutator	#127	C₂×C₈
Frattini, cosocle	#1963	C₂²
prime radical	#532	C₂×C₄
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1963 ◁ G
LECS	G ▷ #1963 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

210 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,10,6,11,28,42,7,13,27,43,30,12,29,44)(3,14,16,45,17,47,53,63,19,48,25,58,18,46,55,60)(4,20,21,41,22,9,36,37,24,38,57,64,23,39,26,59)(8,31,32,15,33,49,61,50,35,52,62,51,34,56,40,54);(1,3,7,19)(2,8,13,35)(4,15,24,51)(5,25,27,16)(6,18,30,17)(9,14,39,48)(10,40,43,61)(11,34,12,33)(20,47,38,46)(21,56,57,49)(22,54,23,50)(26,31,36,52)(28,53,29,55)(32,42,62,44)(37,63,59,60)(41,58,64,45)

Polycyclic presentation:

2^9|a^2=d,b^2=i,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=deh,e^b=eh,f^b=fi,g^b=gh,d^c=dghi,e^c=eh,g^c=gi,e^d=ehi,g^d=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1731);